Math 131: Introduction to Analysis

Fall 2012

Overview

In this course we will begin a study of real analysis, including basic properties if the real numbers, sequences and series, various notions of convergence, differentiation and integration.

Instructors

Dagan Karp
Office hours: Tu/Thu 4:00-5:00 pm, and by appointment.
Office Location: Olin 1267

Tutors and Graders

Rosalie Carlson and Liz Sarapata are tutoring and grading for this course. You should talk to them frequently and take advantage of their expertise! Tutoring hours are:

Sunday: 7-9 pm
Tuesday: 7-9 pm
Tutoring will be held in Sprague 3.

Textbook

The official text for this course is Principles of Mathematical Analysis, by Walter Rudin.

Grades

There will be two midterms, one final exam and weekly homework. Each will count one fifth of your grade. The remaining fifth is the maximum of these four items.

Homework

Written homework will be due in class each Wednesday, and is posted below. Late homework may not be accepted. Each student's lowest homework score will be dropped. Please consult the HMC mathematics homework format guidelines for helpful tips on homework submission and formatting.

Rewrites

Students have the opportunity to turn in a rewritten copy of the first four homeworks for a revised grade. Details can be found on the Math 131 Rewrite Policy.

Exams

The exams will be take home. They will be due Friday Oct. 12, Friday Nov. 9, and Friday Dec. 14 respectively.

LaTeX

Students should feel free to use LaTeX for homework, but this isn't required.

Disabilities

Students who need disability-related accommodations are encouraged to discuss this with the instructor as soon as possible.

Honor Code

Though cooperation on homework assignments is encouraged, students are expected to write up their own solutions individually. That is, no copying. Comprehension is the goal, so even with cooperation, you should understand solutions well enough to write them up yourself. It is appropriate to acknowledge the assistance of others; if you work with others on a homework question, please write their names in the margin. Tests are to be done individually. You are encouraged to discuss your paper with other students. The HMC Honor Code applies in all matters of conduct concerning this course.

Lecture Notes

These lecture notes are generously provided by Julius Elinson. Beware they may contain typo's and mistakes coming directly from the blackboard!

Lecture 2 PDF, TEX
Lecture 3 PDF, TEX
Lecture 4 PDF, TEX
Lecture 5 PDF, TEX
Lecture 6 PDF, TEX
Lecture 7 PDF, TEX
Lecture 8 PDF, TEX
Lecture 9 PDF, TEX
Lecture 10 PDF, TEX
Lecture 11 PDF, TEX
Lecture 12 PDF, TEX
Lecture 13 PDF, TEX
Lecture 14 PDF, TEX
Lecture 15 PDF, TEX
Lecture 16 PDF, TEX
Lecture 17 PDF, TEX
Lecture 18 PDF, TEX
Lecture 19 PDF, TEX
Lecture 20 PDF TEX
Lecture 21 PDF TEX
Lecture 22 PDF, TEX
Lecture 23 PDF, TEX

YouTube Lectures

Francis Su delivered these lectures in 2010. My lectures will be very different, but you may find these valuable for review, or, better yet, watch them before the class lecture, and then during class you can ask questions!

Lecture 1: Constructing the rational numbers
Lecture 2: Properties of Q
Lecture 3: Construction of R
Lecture 4: The Least Upper Bound Property
Lecture 5: Complex Numbers
Lecture 6: The Principle of Induction
Lecture 7: Countable and Uncountable Sets
Lecture 8: Cantor Diagonalization, Metric Spaces
Lecture 9: Limit Points
Lecture 10: Relationship b/t open and closed sets
Lecture 11: Compact Sets
Lecture 12: Relationship b/t compact, closed sets
Lecture 13: Compactness, Heine-Borel Theorem
Lecture 14: Connected Sets, Cantor Sets
Lecture 15: Convergence of Sequences
Lecture 16: Subsequences, Cauchy Sequences
Lecture 17: Complete Spaces
Lecture 18: Series
Lecture 19: Series Convergence Tests
Lecture 20: Functions - Limits and Continuity
Lecture 21: Continuous Functions
Lecture 22: Uniform Continuity
Lecture 23: Discontinuous Functions
Lecture 24: The Derivative, Mean Value Theorem
Lecture 25: Taylor's Theorem
Lecture 26: Ordinal Numbers, Transfinite Induction

Homework Assignments

Homeworks, due Wednesdays in class
All HW's refer Rudin's Principles of Mathematical Analysis.

  • HW #0. Due Monday Sep 10. Read this handout on good mathematical writing and turn in brief answers to these questions.

    • Directly from the handout reading:
      • 1. What is a good rule of thumb for what you should assume of your audience as you write your homework sets?
      • 2. Is blackboard writing formal or informal writing?
      • 3. Do you see why the proof by contradiction on page 3 is not really a proof by contradiction?
      • 4. Name 3 things a lazy writer would do that a good writer wouldn't.
      • 5. What's the difference in meaning between these three phrases?
          "Let A=12."
          "So A=12."
          "A=12."
    • Now examine Section 1.1 of Rudin, showing that there is no rational p that satisfied p2=2.
      • 6. There are many places in his proof where he could have used symbols to express his ideas, but he does not. (e.g., "Let A be the set of all positive rationals p such that...") Why do you think he chooses not to use symbols?
      • 7. What would you change about his presentation if you were writing for a high school audience? Give a specific example.

    Keep in mind the handout and the homework format as you write up your answers.

  • HW #1. Due Sep 12. Read Chapter 1 in its entirety. Do not worry about understanding everything, just read for the big ideas. Turn in the following problems:

    PART ALPHA.

      Problem A. In no more than four sentences, describe the main themes and concepts of Chapter 1. The first sentence or two should be at a level that your parent could understand even if they never went to college. The other sentences should be understandable by any college student.
      Problem B. Recall that in class, we defined a rational number m/n to be an equivalence class of pairs (m,n) under an equivalence relation. We also defined addition of rational numbers in terms of representatives: a/b + c/d = [ad+bc]/[bd]. Show that the addition of rational numbers is well-defined.
      Problem C. Define a multiplication of rational numbers, and show this multiplication is well-defined.

    PART BETA.

      Do also Chapter 1 ( 1, 2, 3a ).

    Please hand in PARTS ALPHA and BETA in two separately stapled parts. (ALPHA and BETA parts will be graded separately.) Each part should have your name and follow this homework format as well as the guidelines for good mathematical writing.

    In all homework, remember to concentrate on good writing.

p.s. Be sure to subscribe to math-131-l (using listkeeper@hmc.edu) if you are not already getting e-mails from the list.

  • HW #2. Due Sep 19. A problem marked "R" means read, but do not do the problem.

    PART ALPHA.

      Chapter 1 ( R3bcd, R4, 5, R7, 8, 9 ).

    PART BETA.

      Chapter 2 ( 2, R3, 4 ) and
      Problem S. For a real number a and non-empty subset of reals B, define: a + B = { a + b : b is in B }. Show that if B is bounded above, then sup( a + B ) = a + sup B.

p.s. Be sure to subscribe to math-131-l (using listkeeper@hmc.edu) if you are not already getting e-mails from the list.

  • HW #3. Due Sep 26.

    Read Chapter 2.

    PART ALPHA.

      Do Chapter 2 ( 5, R6, R7, 8, 11[exclude d2] ).

    PART BETA.

      Do Chapter 2 ( 9ab, 9cd, 9ef, R10 )

  • HW #4. Due Oct 3. A problem marked "R" means read, but do not do the problem.

    PART ALPHA.

      Do Chapter 2 ( 7, 12, R14, 22, R23 ).

    PART BETA.

      Do Chapter 2 ( 16[closed,bounded], 16[not compact, open?], R23, 25 ). Problem 25 requires you to read Problem 23.

  • EXAM 1. Available Monday October 8. Test is due FRIDAY Oct 12 at 3:00pm. You may return it to the Math Department Main Office or slip under my door.

  • HW #5. Due Oct 17. A problem marked "R" means read, but do not do the problem. There are no re-writes from this homework forward.

    PART ALPHA.

      Do Chapter 2 ( R15, 17[countable?, dense?], 17[compact?, perfect?], 18 ).

    PART BETA.

      Do Chapter 2 ( 19, 20, 24, R26 ).

  • HW #6. Due Oct 24. A problem marked "R" means read, but do not do the problem.

    PART ALPHA.

      Do Chapter 3 ( 1, 3, R16, 20 ) Hint on 3.3: can you show the sequence is increasing? Induction may be of help here.

    PART BETA.

      There is no part beta this week. Enjoy Fall Break.

  • HW #7. Due Oct 31. A problem marked "R" means read, but do not do the problem.

    PART ALPHA.

      Do Chapter 3 ( 2, 5, 23 )

    PART BETA.

      Do Chapter 3 ( 24ab, 24cde, 25 )

  • EXAM 2. Due Mon Nov 12.

    You may wish to review your notes in preparation for the midterm.

  • HW #8. Due Nov 14. A problem marked "R" means read, but do not do the problem.

    PART ALPHA.

      Chapter 4 ( 1, 2, 3 )

    PART BETA.

      Do Chapter 4 ( 4, R6, 8 )
    On the reading problem 4.6, you may assume that E is a subset of real numbers and f is a real-valued function, and the distance in R2 (where the graph lives) is the usual Euclidean metric.

  • HW #9. Due Nov 21. A problem marked "R" means read, but do not do the problem.

    PART ALPHA.

      Do Chapter 4 ( 10, 11* ). *You do NOT have do the 2nd part of Problem 4.11, where it says "Use this result to give..."

    PART BETA.

      Do Chapter 4 ( 12, 14, R16, 18, R19 ).

  • HW #10. Due Nov 28.

    PART ALPHA.

      Do Problem E: Use the mean value theorem to show that e^x is greater than or equal to (1+x) for all x in R. (You may assume knowledge of the derivative of e^x.)

      and Chapter 5 ( 1, 4 ).

    PART BETA.

      Chapter 5 ( 13ab, R13cdefg, R25abd, 25c ) and Chapter 7 ( R2, R3, 7 ). In Problem 5.13, there is a typo (in some editions of the book): the xa should say |x|a.

  • HW #11. Due Dec 5.

    PART ALPHA.

      Do Chapter 6 ( 1, 2, 4 ).

    Exam 3.
    Out Dec 7.
    Due Dec 14.

Possible upcoming homeworks

Below this line, all homeworks are TENTATIVE. This means they are likely to be assigned, but there is no guarantee that they will until you see them moved to the box ABOVE. I am putting them here in case you want to work ahead!